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Question 30020: Someone please help me solve these problems. Thanks!
1. Geometry. The area of a rectangle of length x is given by 3x^2+5x. Find the width of the rectangle.
Find all positive values for k for which each of the following can be factored.
2. x^2+x-5
Determine whether the following trinomials is a perfect square. If it is, factor the trinomial.
3. x^2-24x+48
Rewrite the middle term as the sum of two terms and then factor completely.
4. 12w^2+19w+4
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! 1) Geometry. The area of a rectangle of length x is given by 3x^2+5x. Find the width of the rectangle.
Given the length of the rectangle = x
Let the width of the rectangle be y.
The area of the rectangle = 3x^2+5x
That is (length X width) = 3x^2+5x
That is (xXy) = x(3x+5)
Dividing both the sides by x
(xy)/x = x(3x+5)/x
That is y = (3x+5) (cacelling x)
Therefore width of the rectangle = (3x+5) units where x is the given length.
2) x^2+x-5
=(1/4)X (4x^2+4x-20) (multiplying and dividing by)
=(1/4)[(4x^2+4x +1-1-20)]
=(1/4)[(4x^2+4x+1)-21]
=(1/4)[(2x+1)^2-21]
=(1/4)[A^2-B^2] (where A =(2x+1) and B = sqrt(21) )
=(1/4)[(A+B)(A-B)]
=(1/4)[(2x+1)+ rt(21)]X[(2x+1)- rt(21)]
=(1/4){(2x)+[1+rt(21)]}{(2x)+[1-rt(21)]}
Note: Why did we multiply and divide by 4?
Ans: To create a perfect square to make factoring easier!
3. x^2-24x+48
The first term x^2 is a perfect square and is the square of x
That is x^2 = (x)^2
Observe that the constant term is 48
And 48 = (4^2)X3 = [4X(rt3)]^2
Observe that the middle term, that is the term in x is negative
If the given expression were to be a perfect square then the middle term
(-24x) should be -2X(sqrt of the first term)X(sqrt of the constant term)
= - 2x X [4X(rt3)]= -8x(rt3)
Since (-24x) NOT EQUAL TO (-8x(sqrt3))
the given expression is not a perfect square
Note: If the given expression is a perfect square then the free constant in this case should be (12)^2 = 144 so that
x^2-24x+ 144 = (x-12)^2
Note: If the first two terms are of the form (x^2-2xy)
then (y^2) should be the required term to make it a perfect square.
So x^2-24x = (x)^2- 2X(x)X(12) and hence (12)^2 = 144 is required
to make it a perfect square.
4) 12w^2+19w+4
= 12w^2 +(16w + 3w)+4
=4w(3w+4)+1(3w+4)
=4wp+p where p= (3w+4)
=p(4w+1)
=(3w+4)(4w+1)
Note:The product of the square term and the constant term is
(12w^2)X(4) = 48w^2 = (1X2X2X2X2X3)w^2 = (2X2X2X2w)X(3w)
So that 19w = (16w+3w) and (16w)X(3w) = 48w^2
Since the product is positive and the middle term is also positive,both the parts should be positive so that positive added to positive is positive and positive multiplied by positive is positive
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